Noise-stability and central limit theorems for effective resistance of random electric networks

Abstract

We investigate the (generalized) Walsh decomposition of pointto-point effective resistances on countable random electric networks with i.i.d. resistances. We show that it is concentrated on low levels, and thus point-to-point effective resistances are uniformly stable to noise. For graphs that satisfy some homogeneity property, we show in addition that it is concentrated on sets of small diameter. As a consequence, we compute the right order of the variance and prove a central limit theorem for the effective resistance through the discrete torus of side length n in Z d , when n goes to infinity.Proof. Let us first suppose that G is finite. Then it is well known that i u,v r (e) and R u,v (r) are rational functions of r with no positive pole.

The additive structure of elliptic homogenization

“…Versions of (11.3) have been proved in [34,37,8,35,18], with the quantity J replaced by spatial averages of the energy density of the correctors and approximations to the corrector. A version of (11.5) was proved in [31,30], while a version of (11.6) with u r (z) replaced by ∫ Φz,r (u − E[u]) was proved in [24].…”

Quantitative estimates on the periodic approximation of the corrector in stochastic homogenization

“…More recent results go beyond variance estimates and address statistical properties of solutions such as central limit theorems (CLT). In particular, for the approximation of homogenized coefficients, a CLT was recently obtained by Biskup, Salvi and Wolff [1] in the case of small ellipticity ratio (that is, for a coefficient field a close to identity) for independent and identically distributed coefficients, see also Rossignol [15]. An optimal quantitative version of this CLT (which estimates in particular the Wasserstein distance to normality) is proved by Nolen and the first author in [4] (without the smallness assumption on the ellipticity ratio).…”

Effective multipoles in random media

“…Since this property also holds for X h m /X h m−2 and W h m ⊕ W h m−1 , the isomorphism of W h m ⊕ W h m−1 and W m ⊕ W m−1 constructed above lifts to the desired isomorphism of the normed linear spaces X h m /X h m−2 and X m /X m−2 . Finally, we argue that the relation u h ↔ u provided by the isomorphism of X h m /X h m−2 and X m /X m−2 constructed above coincides with the one defined through the qualitative relation (44). By construction, the relation u h ↔ u defined through the isomorphism has the following characterizing property: There…”

“…As in Section 7, we haveũ h = 0 for m = 2. We are therefore in the position to appeal to Theorem 3. We now claim that the relation X h m ∋ u h ↔ u ∈ X m given by (17) is equivalent to the one given by condition (44) (45). Since by Theorem 3 this last two conditions define an isomorphism between X h m /X h m−2 and X m /X m−2 and between Y h m−1 /Y h m+1 and Y m−1 /Y m+1 which preserves the bilinear form, this is enough to conclude the proof of the theorem.…”

“…This was extended by Gu and Mourrat, in a non-obvious way, to characterize the leading order of the fluctuations of the homogenization error and to show that they are Gaussian [36], still relying on [39,Theorem 1]. Incidentally, the Gaussianity of fluctuations in stochastic homogenization was first established on the level of the error in the representative volume element method: [18] in the smallcontrast case, [30] based on Nolen's [42], and [44]. A quite different approach to, among other things, Gaussianity of leading-order fluctuations of the corrector, motivated by the approach to quantitative stochastic homogenization in [7], was carried out by Armstrong, Kuusi and Mourrat [5] (after being announced in [36]).…”